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Domain-Decomposition Methods 255
n En
12 .557106
16 .4418 × 10 −1
20 .4262 × 10 −2
24 .6324 × 10 −3
28 .4265 × 10 −4
32 .7923 × 10 −5
Figure 11.2.1 - Plot of the Table 11.2.1 - Errors for a single-domain
function U(x), x ∈ [−1,1]. approximation of problem (11.2.1).
In the second experiment, we consider two domains, namely S1 =]s0,s1[=] − 1,0[ and
S2 =]s1,s2[=]0,1[. In each interval Sk, 1 ≤ k ≤ 2, we approximate the solution by a
polynomial pnk,k ∈ Pnk , with the collocation method at the Legendre nodes. At the
point s1 = 0 we impose pn1,1(s1) = pn2,2(s1) and condition (11.2.12) for m = 2. We
report in table 11.2.2 the error En1,n2 :=
2
k=1
for various choices of the parameters n1 and n2.
nk
j=1 (pnk,k − U) 2 (θ (nk,k)
j
n1 En1,2 En1,4 En1,6 En1,8
) ˜w (nk,k)
j
12 .130794 .1817 × 10 −1 .1839 × 10 −1 .1839 × 10 −1
16 .133438 .1645 × 10 −2 .1022 × 10 −2 .1021 × 10 −2
20 .133418 .1274 × 10 −2 .5449 × 10 −4 .2638 × 10 −4
24 .133418 .1274 × 10 −2 .4772 × 10 −4 .2089 × 10 −5
28 .133418 .1274 × 10 −2 .4769 × 10 −4 .9319 × 10 −6
Table 11.2.2 - Errors for a multidomain approximation of problem (11.2.1).
1
2